Fechnerian scaling: Metric from discriminability

Purdue University, West Lafayette IN

Investigators

Abstract

This project elaborates and expands the theory and applications of Multidimensional Fechnerian Scaling (MDFS). Intuitively, MDFS is about how to compute distances among objects "from the point of view" of a perceiver (human observer, technical system, group of people, neurophysiological system), based on the probabilities with which this perceiver discriminates (tells apart) very similar objects. Discrimination probabilities impose on the set of perceived objects a local geometric structure (generalized Finsler geometry), and this local structure can be extracted and expanded into a global metric. Essentially, this is an idea with which, albeit in a very limited context, G.T. Fechner launched scientific psychology some 150 years ago. MDFS is motivated by the expectation that, the discrimination among stimuli being arguably the most basic ability of a perceiving system, and the probability of discrimination being a universal measure of discriminability, distances computed from discrimination probabilities may have a fundamental status among social and behavioral measurements. Originally developed for sets of continuously parametrized stimuli (such as colors, sounds, or mixtures of food ingredients) and critically based on computation of derivatives, in the present project MDFS is expanded to discrete sets of objects, such as alphabets, words, or consumer products. When a discrete set of objects is subjected to the proposed modification of MDFS, it is transformed into a network with inter-object distances. A subsequent immersion of this network in a Euclidean space (by means of conventional metric multidimensional scaling) allows one to identify the relevant features that determine the dissimilarities among the objects. The theoretical work in this part of the project will be complemented and guided by the collection of large data sets on discrimination probabilities for such objects as letters of familiar and unfamiliar alphabets and schematic faces. The adaptation of MDFS to discrete object spaces is a non-trivial enterprise that is only enabled by the recent discovery and documentation of two basic properties of discrimination probabilities (regular minimality and nonconstant self-similarity). Their further experimental analysis constitutes the second part of the project. These two properties have far reaching consequences both within and without MDFS. Thus it has been shown that no "random utility" type model in which random representations depend on objects sufficiently smoothly can account for these properties (the models referred to are those in which objects are mapped into random entities in some unobservable "internal" space, and the decision as to whether the objects are the same or different is determined by the realizations of these random entities). The third part of the project is aimed at an in-depth analysis of the relationship between MDFS, the "random utility" type models, and the recently proposed alternative to such models in which random entities are replaced by "uncertainty blobs" of perceived objects. A successful completion of the project will improve our understanding of the fundamental notions of discrimination and dissimilarity, and it will advance the foundations of measurement in social and behavioral sciences. The project also has a clear applied focus. Thus, the development of algorithms and software for MDFS in discrete object spaces will provide a new useful tool for analyzing large-scale polls of public opinion, for consumer surveys, and for educational testing.